Hybrid form of quantum theory with non-Hermitian Hamiltonians

Abstract

In 1956, Freeman Dyson discovered that the practical solution of Schrödinger equation may be facilitated when one replaces the standard self-adjoint Hamiltonian $\mathfrak{h}={\mathfrak{h}}^{†}$ by its manifestly non-Hermitian isospectral avatar $H={\Omega }^{-1}\mathfrak{h}\Omega$ with ${\Omega }^{†}\Omega =\Theta \ne I$. The standard unitary interpretation of the evolution can be then achieved either by the ``operator transformation'' (OT) reconstruction of $\mathfrak{h}$, or via a metric-amendment (MA) change of the Hilbert space, ${\mathcal{H}}_{auxiliary}\to {\mathcal{H}}_{physical}$. In our present letter we describe an alternative Hermitization of a ``hybrid form'' (HF) which simply combines the OT model-building strategy with the MA model-building strategy. The merits of the approach are illustrated using a schematic two-state quantum system.

15 pages

References in corpus (12)

• Making Sense of Non-Hermitian Hamiltonians
• An Equivalent Hermitian Hamiltonian for the non-Hermitian -x^4 Potential
• Time-dependent version of cryptohermitian quantum theory
• Three-Hilbert-Space Formulation of Quantum Mechanics
• Exact analytical solutions for time-dependent Hermitian Hamiltonian systems from static unobservable non-Hermitian Hamiltonians
• Flattening the Curve with Einstein's Quantum Elevator: Hermitization of Non-Hermitian Hamiltonians via a Generalized Vielbein Formalism
• Non-Hermitian Heisenberg representation
• Spectrally equivalent time-dependent double wells and unstable anharmonic oscillators
• Wheeler-DeWitt equation and the applicability of crypto-Hermitian interaction representation in quantum cosmology
• Dynamics of Disordered Mechanical Systems with Large Connectivity, Free Probability Theory, and Quasi-Hermitian Random Matrices
• Feasibility and method of multi-step Hermitization of crypto-Hermitian quantum Hamiltonians
• Solvable quantum lattices with nonlocal non-Hermitian endpoint interactions

Cited by in corpus (9)

• Composite quantum Coriolis forces
• Quasi-Hermitian formulation of quantum mechanics using two conjugate Schrödinger equations
• Non-Hermitian Generalization of Rayleigh-Schrödinger Perturbation Theory
• Non-stationary quantum mechanics in hybrid non-Hermitian interaction representation
• Intrinsic exceptional point -- a challenge in quantum theory
• Phase transitions in quasi-Hermitian quantum models at exceptional points of order four
• Three alternative model-building strategies using quasi-Hermitian time-dependent observables
• Bosonic Holes in Quadratic Bosonic Systems
• On the $\mathcal{P}\mathcal{T}$-symmetric parametric amplifier